$f(x)=x(4 x+9)(x-2)(2 x-9)(x+5)$ has zeros at $x=-5, x=-\frac{9}{4}, x=0, x=2$ , and $x=\frac{9}{2}$ . $\newline$ What is the sign of $f$ on the interval $0<x<2$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $f$ is always positive on the interval. $\newline$ (B) $f$ is always negative on the interval. $\newline$ (C) $f$ is sometimes positive and sometimes negative on the interval.

Full solution

f(x)=x(4 x+9)(x-2)(2 x-9)(x+5)

has zeros at

x=-5, x=-\frac{9}{4}, x=0, x=2

, and

x=\frac{9}{2}

\newline

What is the sign of

f

on the interval

0<x<2

\newline

Choose

1

answer:

\newline

(A)

f

is always positive on the interval.

\newline

(B)

f

is always negative on the interval.

\newline

(C)

f

is sometimes positive and sometimes negative on the interval.

Factors Sign Analysis: Since $f(x)$ is a product of factors, the sign of $f(x)$ on the interval $0 < x < 2$ depends on the sign of each factor in that interval.
Consideration of Zeros: The zero at $x=0$ doesn't affect the interval $0 < x < 2$ , because we're looking at values of $x$ that are greater than $0$ .
Endpoint Consideration: The zero at $x=2$ is the endpoint of the interval, so we only consider values of $x$ that are less than $2$ .
Positive Factors: The factors $(4x+9)$ , $(x-2)$ , $(2x-9)$ , and $(x+5)$ are all positive for values of $x$ in the interval $0 < x < 2$ , because their zeros are outside this interval.
Conclusion: Since all factors are positive in the interval $0 < x < 2$ , the function $f(x)$ is always positive on this interval.